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Thermal rectification in a bilayer wall: Coupled radiation and conduction heat transfer
Applied Thermal Engineering 107 (2016) 1248–1252
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Thermal rectification in a bilayer wall: Coupled radiation and conduction heat transfer Hamou Sadat ⇑, Vital Le Dez Institut P’, Université de Poitiers, Centre National de la Recherche Scientifique, 2 Rue Pierre Brousse, Bâtiment B25, TSA 41105, 86073 Poitiers Cedex 9, France
h i g h l i g h t s Coupled conduction and radiation bilayer walls can be used as thermal diodes. Rosseland and thin media approximations lead to equal maximum rectification factors. Rectification factor of 2.62 is theoretically possible.
a r t i c l e
i n f o
Article history: Received 16 March 2016 Revised 8 June 2016 Accepted 13 July 2016 Available online 14 July 2016 Keywords: Rectification factor Diode effect Conduction Radiation
a b s t r a c t Like electronic diode developed for the control of electric current, it may be desirable to build thermal diodes that control heat transfer. This thermal rectification effect can occur when heat is transferred asymmetrically. We consider in this work, the case of a composite wall, one layer of which is both conductive and radiative. Both Rosseland and surface radiation approximations are used for taking into account radiative transfer. Analytical expressions for the rectification factor are given. It is shown that this factor is bounded by a maximum value of around 2.62. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Thermal rectification which occurs when forward heat fluxes are greater than reverse counterparts has been the subject of numerous studies because of the potential applications in heat management [1]. This phenomenon which has been first described in [2] can appear in heat radiation, heat convection and heat conduction as well. Natural convection in a fluid between two horizontal plates with imposed temperatures is a good example. If the upper plate is hotter than the lower, heat is transferred mainly by conduction. If the lower plate is hotter, natural convection may take place and the Nusselt number is much greater. As heat flow in one direction is much higher than in the opposite direction, this kind of device has been called thermal diode and this effect thermal rectification. Radiative thermal rectifiers have been studied in near field and far field radiation. Rectification can be achieved in a device with two selective emitters one of which has radiative properties strongly dependent on temperature
⇑ Corresponding author. E-mail address: [email protected] (H. Sadat). http://dx.doi.org/10.1016/j.applthermaleng.2016.07.082 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
whereas the other one has temperature independent properties [3]. In order to enhance the rectification ratio, phase change radiative thermal rectifier based on the phase transition of insulatormetal transition materials around the operating temperature has also been considered [4]. The rectification is realized by the phase-change of vanadium dioxide (VO2), which behaves as metal at high temperature (>340 K) and insulator at low temperature (<340 K). Convective thermal diode walls where the heat flow is more intense in one direction than in the other has been studied in [5–8] where rectification factors as high as 8 have been reported. The idea of heat conduction thermal rectifier in a composite wall consisting of two materials, each having temperature dependant conductivity was presented in [9,10]. Some experimental results have also been discussed [11–13]. More recently, it has been shown [14] that the maximum rectification factor in a composite wall consisting of two solid materials, each having thermal conductivities with different temperature linear dependences, is equal to 3. In this paper we focus on steady coupled heat conduction and radiation problem within a one dimensional wall composed of two different materials. The first material is radiative and conductive while the second is only conductive and each thermal conductivity
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H. Sadat, V. Le Dez / Applied Thermal Engineering 107 (2016) 1248–1252
Nomenclature k1, k2 qt q+, q S T Tc Tm R x q, r, s t, u
thermal conductivities of the two considered media (W m1 K1) total heat flux in the medium (W m2) heat flux of the forward and reverse cases (W m2) volumic radiative source (W m3) temperature field in the medium (K) temperature of the interface between medium 1 and 2 (K) normalization temperature (K) rectification factor coordinate axis direction dimensioned coefficients of the forward and reverse 4th degree equations of the interface temperature solutions of the forward and reverse resolvant 3rd degree equations
is constant. We shall assume that radiative transfer can be described by the Rosseland approximation and therefore has a power three increase of its radiative conductivity with temperature. This variation of the total conductivity of the first wall with temperature let us expect a rectification factor of the order of that obtained in pure heat conduction problems. In the following sections, the problem is solved analytically and an expression for the rectification factor is given. It is shown that the rectification factor has a maximum value of 2.6 which is indeed of the order of what is obtained in pure heat conduction problems. In order to examine the influence of the extinction coefficient, we have also considered the limiting case of an optically thin media where only surface radiation is taken into account in the first layer. Here again, it is found that rectification factor is bounded by the same maximum value. 2. Heat conduction model and radiation Rosseland approximation Let us consider a one-dimensional heat exchange problem with two solid different materials as depicted in Fig. 1. The first domain is a conductive hot emitting and highly absorbing grey semitransparent medium, characterized by its thermal conductivity k1 and its absorption coefficient j, while the other domain is made of an opaque conductive medium characterized by its thermal conductivity k2 only. The non-dimensional width of each of them is 0.5 giving a total thickness of 1. The thermal conductivities k1 and k2 of the two materials are both supposed to be constant. The first material is a participating medium with a high absorption coefficient, for which radiative transfer within it can be described by the Rosseland approximation. This approximation stands that the radiative transfer can be described as a conductive one, with an apparent thermal conductivity depending both on the absorption coefficient and the temperature. Its total conductivity can therer T 3 is the fore be written as: kðTÞ ¼ k1 þ kR ðTÞ, where kR ðTÞ ¼ 16 3j Rosseland conductivity. Without loss of generality we will suppose that the boundary conditions are T(0) = T1 and T(1) = 0. The problem with an imposed non zero temperature at x = 1 can be retrieved by a variable change. These boundary conditions which
y, z
non dimensional solutions of the forward and reverse resolvant equations
Greek letters a, b, c non-dimensional coefficients analogous to q, r, s j absorption coefficient (m1) k total conductivity (W m1 K1) r Stephan-Boltzmann constant (5.67 108 W m2 K4) q, s non dimensional variables defining the rectification factor q1, qc reflection factors of the boundary surfaces of medium 1 e1, ec emissivities of the boundary surfaces of medium 1
correspond to the forward case are inverted in the reverse case (T(0) = 0 and T(1) = T1). In all what follows, one notes k for the total conductivity: in the opaque medium 2 it reduces only to the thermal conductivity k2, while in the radiatively participating medium 2 it is the sum of the thermal conductivity k1 and the Rosseland conductivity kR(T). The heat equation writes everywhere in the slab:
d dT ¼0 k dx dx
ð1Þ
The solution is simply given by k dT ¼ b where b is a constant dx depending on the boundary conditions. 2.1. Forward case In the second domain x 2 12 ; 1 where k ¼ k2 , the differential equation for the temperature field has the simple general solution T ¼ kb2 x þ c, c being another constant to be determined, associated to the two boundary conditions T(1) = 0 and T 12 ¼ T c . The determination of the two unknown constants easily gives the heat flux field in the domain: qt = 2k2Tc. In the first region: x 2 0; 12 , we can write:
k
dT d 4r 4 T ¼ k1 Tþ dx dx 3jk1 The
heat
equation
has
ð2Þ therefore
the
general
solution
T þ 34jrk1 T 4 ¼ kb1 x þ c. The two boundary conditions T(0) = T1 and T 12 ¼ T c allow the determination of the two constants, which leads to the heat flux:
4r 3 4r 3 qt ¼ 2k1 T c 1 þ Tc T1 1 þ T1 3jk1 3jk1
ð3Þ
The continuity of the heat flux at x ¼ 12 gives the algebraic equation verified by the interface temperature:
T 4c þ
3jk1 k2 3jk1 T 1 4r 3 Tc 1þ 1þ T1 ¼ 0 4r k1 4r 3jk1
Fig. 1. Heat conduction problem.
ð4Þ
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This 4th degree equation, of general form T 4c ¼ qT c þ r, with
q ¼ 34jrk1 1 þ kk21 and r ¼ 3j4kr1 T 1 1 þ 34jrk1 T 31 , can exactly be solved by the Ferrari’s method, and the temperature Tc at the interface between the two media can be expressed as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffi 1 2jqj pffiffi t t Tc ¼ 2 t
where t ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 q2 þ q4 þ4ð4r 3Þ 2
sq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 3 q4 þ4ð4r q2 3Þ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffi 2jqj pffiffi t t t
s
ð5Þ
Ts is a reference temperature, and that the temperature Tm can be seen as the reference temperature for which the corresponding Stark number equals 13. With these notations, the coefficients of the 4th degree interface temperature equations for the forward and reverse cases define 3
> 0 is the solution of the
non-dimensional coefficients a, b and c, with jqj ¼ aT 3m , r ¼ bT 4m
third degree associated resolvant equation t3 + 4rt q2 = 0. Then the heat flux of the forward case is simply:
qþ ¼ k2
Note that for a steady-state conduction-radiation transfer inside a semi-transparent grey medium of absorption coefficient j and thermal conductivity k, one may introduce a conductionradiation parameter, the Stark number, defined by N S ¼ 4rjTk 3 , where
ð6Þ
and s ¼ cT 4m , where a = 1 + q, b = s(1 + s3) and c = qs. Introducing sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 a2 þ a4 þ4ð4b 3Þ the two supplementary quantities: z¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sq sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 3 3 3 3 a4 þ4ð4b a2 a2 þ a4 þ4ð43cÞ a4 þ4ð43cÞ a2 3Þ and y ¼ , with 2 2 2 z ¼ Tt2 and y ¼ Tu2 , which are the non-dimensional Cardan solutions m
2.2. Reverse case In the reverse case, the general solution obtained in the forward case remains valid, with T ¼ kb2 x þ c for x 2 12 ; 1 and 0 T þ 34jrk1 T 4 ¼ kb1 x þ c0 for x 2 0; 12 . Application of the boundary conditions sets leads to the two expressions of the heat flux as qt = 2k2(T1 Tc) in the 2nd region
and qt ¼ 2k1 T c 1 þ 34jrk1 T 3c in the first one. Similarly to what precedes, the heat flux continuity gives the algebraic 4th degree equation verified by the interface temperature under the form:
T 4c þ
3jk1 k2 3jk2 T 1 Tc 1þ ¼0 4r k1 4r
ð7Þ
Defining the parameter s ¼ 3j4kr2 T 1 gives the general form of the 4th degree equation for the interface temperature T 4c ¼ qT c þ s, where the parameter q has the same expression as in the forward case, and the solution u of the associated resolvant third degree sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 3 3 q2 þ q4 þ4ð4s q4 þ4ð4s q2 3Þ 3Þ > 0. The temperature equation: u ¼ 2 2 Tc at the interface between the two media and the heat flux of the reverse case are finally given by the following expressions:
1 Tc ¼ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi 2jqj pffiffiffi u u u
ð8Þ
q ¼ k2
qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2affi p z z z Rðq; sÞ ¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2s þ y p2affiffiy y
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi 2jqj pffiffiffi u u 2T 1 u
ð9Þ
2.3. Rectification factor The steady state heat fluxes in the forward and the reverse cases given by relations (6) and (9) allow us to define the rectification
þ
factor as the ratio R ¼ qq . This factor is given by the following
ð11Þ
It is possible to show by taking the limit of the previous expression that R ? 1 for s ? 0. To obtain this result, a limited develop qffiffiffiffiffiffiffiffiffiffiffiffi 4 1 2affi p zs!0 a3 1 þ 4a3 3 s , ment in the variable s shows that z qffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffi 4 1 1 3 2affiffi p zs!0 a3 1 2a3 3 s , ys!0 a3 1 þ 4qa3 s and y q ffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffi pffiffiffi 1 3 ys!0 a3 1 2qa3 s , whence 2s þ y p2affiffiy ys!0 qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2affi 2s 1 qa ¼ 2as and p z zs!0 2as for any q. z
We present in Fig. 2 the rectification factor as a function of q and s. The first map on Fig. 2a shows the general representation of the rectification factor for a large spectrum of the two parameters q and s, while the Fig. 2b focuses on a smaller region of small s where the rectification factor takes significant values >1. As it can be observed, there exists a thin band whose thickness increases with r where R has high values. The maximum value of the rectification factor is numerically seen to be around: Rm 2.62. The observation of the numerical results shows that if q P 3 then
R P 2 if 0:73q3 6 s 6 1:47q3 . In particular, it can be seen that R 6 1:5 as soon as s P 15 for reasonable q, and that R ? 1 for s ? + 1 for any q. As a matter of example, one can see that a value of the rectification factor R around 2 can be reached only for s P 1:3 which implies that the chosen temperature T1 must be high enough if the absorption coefficient is not too small: if one chooses T1 = 1000 K, one obtains Tm 770 K and jk1 34:5. This case corresponds to a high absorption coefficient and a small thermal conductivity k1 with a large spectrum of conductivity of the second layer: k2 P 2k1 . 1
and
m
of the two resolvant third degree equations of the forward and reverse cases, allows writing the rectification factor as a function of the two independent non-dimensional parameters q and s under the form:
1
formula:
qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi 2jqj pffi t t t R¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi u 2T 1 þ u 2jqj u
ð10Þ
Let us define a normalization temperature as T 3m ¼ 34jrk1 which appears in Eqs. (4) and (7), and the two non-dimensional parameters
s ¼ TTm1 and q ¼ kk21 .
3. Heat conduction model and thin media radiation approximation In the case of a thin radiation approximation with an absorption coefficient tending towards 0 (surface radiation), conservation of energy in medium 1 leads to write that the total heat flux qt = qr + qc, sum of the radiative flux and the conductive one, is constant across the medium. This implies that:
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Fig. 2. Rectification factor versus the two parameters q and s.
re1 ec ðT 41 T 4c Þ dT ¼x k1 1 q1 qc dx
ð12Þ
where x is a constant depending on the boundary conditions.
T 4c þ
3.1. Forward case In the first domain x 2 0; 12 , Eq. (12) can be solved as follows:
1 TðxÞ ¼ k1
"
#
re1 ec ðT 4i T 4c Þ x xþb 1 q1 qc
ð13Þ
where b is a constant to be determined. The boundary conditions, re e ðT 4 T 4 Þ T 12 ¼ T c and T(0) = T1, easily lead to: x ¼ 11c q 1q c 2k1 ðT c T 1 Þ, 1
c
and the total flux in domain 1 writes:
qt ¼
re e
4 1 c ðT 1
T 4c Þ
1 q1 qc
2k1 ðT c T 1 Þ
ð14Þ
There is no modification to the previous case in the second domain x 2 12 ; 1 where the total flux consists only in a conductive one, given by qt = 2k2Tc. Continuity of the heat flux at the interface leads to the 4th degree equation verified by the temperature Tc:
k2 Tc 1þ e1 ec r k1 2k1 ð1 q1 qc ÞT 1 e1 ec r 1þ T 31 ¼ 0 e1 ec r 2k1 ð1 q1 qc Þ
T 4c þ
2k1 ð1 q1 qc Þ
ð15Þ
which is exactly similar to the equation obtained in the Rosseland approximation, with the substitution:
q1 qc Þ j $ 8ð1 . 3e1 ec
the
first
domain
x 2 0; 12 ,
the
total
flux
writes:
x ¼ r1e1qe1c qT cc k1 dT , and the temperature variation is given by: dx 4
! 1 re1 ec T 4c TðxÞ ¼ þx xþb k1 1 q1 qc The boundary conditions, T total flux expression:
qt ¼ 2k1 T c 1 þ
2k1 ð1 q1 qc Þ
e1 ec r
k2 2k2 ð1 q1 qc ÞT 1 Tc 1þ ¼0 k1 e1 ec r
The same substitution
ð18Þ
q1 qc Þ j $ 8ð1 allows to retrieve the equa3e1 ec
tion verified in the Rosseland approximation. q1 qc Þ Then introducing the normalization temperature T 3m ¼ 2k1 ð1 e1 ec r ,
so as the independent parameters s ¼ TTm1 and q ¼ kk21 , allows writing the rectification factor as:
qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2affiffi p z z x R¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2s þ y p2affiffiy y
ð19Þ
where a, z and y have exactly the same expressions as in the Rosseland approximation frame. One concludes that the map representing the rectification factor for the thin media approximation is identical to the one obtained for the Rosseland approximation, and that the maximal value of the rectification factor is Rm 2.62. In the case of black boundaries, the normalization temperature is: T 3m ¼ 2kr1 . For a thin medium with k1 = 1 W m1 K1, this leads to Tm 328 K and s 3 when T1 = 1000 K and relatively high values of R around 2 can be obtained for media with k2 P 2, the maximum Rm being reached for k2 30. If k1 ¼ 0:1 W m1 K1 , Tm 152 K and s 6.6, leading to the maximal value of R reached for media with k2 P 18. 4. Conclusion
3.2. Reverse case In
Similarly there is no modification in the area 12 6 x 6 1 where the flux writes qt = 2k2(T1 Tc). The equation verified by the interface temperature Tc is:
1 2
e1 ec r T3 2k1 ð1 q1 qc Þ c
ð16Þ ¼ T c and (0) = 0, now lead to the
ð17Þ
In this study we developed an analytical expression for the rectification factor in a bilayer wall when radiation and conduction are coupled in the first layer. By using the Rosseland approximation and by considering surface radiation in the first layer where conduction is also taken into account while the second layer is only conductive, we have given an analytical expression for the rectification factor. A theoretical maximum value has been numerically found, which can hardly be obtained in the Rosseland approximation frame, except for a very small thermal conductivity in the hot domain. On the contrary for the thin media approximation, values close to the theoretical maximum rectification factor can be obtained in less restrictive conditions.
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References [1] N.A. Roberts, D.G. Walker, A review of thermal rectification observations and models in solid materials, Int. J. Thermal Sci. 50 (2011) 648–662. [2] C. Starr, The copper oxide rectifier, J. Appl. Phys. 7 (1935) 15–19. [3] Svend-Age Biehs 1, Philippe Ben-Abdallah 2, Felipe S.S. Rosa, Nanoscale radiative heat transfer and its applications, in: Vasyl Morozhenko (Ed.), Infrared Radiation, InTech, 2012. ISBN: 978-953-51-0060-7. [4] Philippe Ben-Abdallah, Svend-Age Biehs, Phase-change radiative thermal diode, Appl. Phys. Lett. 103 (2013) 191907. [5] H. Nakamura, Y. Asako, Heat transfer in a parallelogram shaped enclosure, Bull. JSME 23 (185) (1980). [6] A. Reis, I. Smith, Convective thermal rectification in an air-filled parallelogramic cavity, 1987. [7] N. Seki, S. Fukusako, A. Yamaguchi, An experimental study of free convective heat transfer in a parallelogrammic enclosure, J. Heat Transfer 105 (August) (1983).
[8] J.M. Hyun, B.S. Choi, Transient natural convection in a parallelogram-shaped enclosure, Int. J. Heat Fluid Flow 11 (2) (1990). [9] M. Terraneo, M. Peyrard, G. Casati, Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier, Phys. Rev. Lett. 88 (9) (2002) (0943021-4). [10] M. Peyrard, The design of a thermal rectifier, Europhys. Lett. 76 (2006) 49. [11] W. Kobayashi, Y. Teraoka, I. Terasaki, An oxide thermal rectifier, Appl. Phys. Lett. 95 (2009) 171905. [12] B. Hu, D. He, L. Yang, Y. Zhang, Thermal rectifying effect in macroscopic size, Phys. Rev. E 74 (2006) 060201. [13] Wataru Kobayashi, Daisuke Sawaki, Tsubasa Omura, Takuro Katsufuji, Yutaka Moritomo, Ichiro Terasaki, Thermal rectification in the vicinity of a structural phase transition, Appl. Phys. Express 5 (2) (2012). [14] Tien-Mo Shih, Zhaojing Gao, Ziquan Guo, Holger Merlitz, Patrick J. Pagni, Zhong Chen, Maximal rectification ratios for idealized bi-segment thermal rectifiers, Sci. Rep. 5 (2015) 12677, http://dx.doi.org/10.1038/ srep12677.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Thermal rectification in a bilayer wall: Coupled radiation and conduction heat transfer Hamou Sadat ⇑, Vital Le Dez Institut P’, Université de Poitiers, Centre National de la Recherche Scientifique, 2 Rue Pierre Brousse, Bâtiment B25, TSA 41105, 86073 Poitiers Cedex 9, France
h i g h l i g h t s Coupled conduction and radiation bilayer walls can be used as thermal diodes. Rosseland and thin media approximations lead to equal maximum rectification factors. Rectification factor of 2.62 is theoretically possible.
a r t i c l e
i n f o
Article history: Received 16 March 2016 Revised 8 June 2016 Accepted 13 July 2016 Available online 14 July 2016 Keywords: Rectification factor Diode effect Conduction Radiation
a b s t r a c t Like electronic diode developed for the control of electric current, it may be desirable to build thermal diodes that control heat transfer. This thermal rectification effect can occur when heat is transferred asymmetrically. We consider in this work, the case of a composite wall, one layer of which is both conductive and radiative. Both Rosseland and surface radiation approximations are used for taking into account radiative transfer. Analytical expressions for the rectification factor are given. It is shown that this factor is bounded by a maximum value of around 2.62. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Thermal rectification which occurs when forward heat fluxes are greater than reverse counterparts has been the subject of numerous studies because of the potential applications in heat management [1]. This phenomenon which has been first described in [2] can appear in heat radiation, heat convection and heat conduction as well. Natural convection in a fluid between two horizontal plates with imposed temperatures is a good example. If the upper plate is hotter than the lower, heat is transferred mainly by conduction. If the lower plate is hotter, natural convection may take place and the Nusselt number is much greater. As heat flow in one direction is much higher than in the opposite direction, this kind of device has been called thermal diode and this effect thermal rectification. Radiative thermal rectifiers have been studied in near field and far field radiation. Rectification can be achieved in a device with two selective emitters one of which has radiative properties strongly dependent on temperature
⇑ Corresponding author. E-mail address: [email protected] (H. Sadat). http://dx.doi.org/10.1016/j.applthermaleng.2016.07.082 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
whereas the other one has temperature independent properties [3]. In order to enhance the rectification ratio, phase change radiative thermal rectifier based on the phase transition of insulatormetal transition materials around the operating temperature has also been considered [4]. The rectification is realized by the phase-change of vanadium dioxide (VO2), which behaves as metal at high temperature (>340 K) and insulator at low temperature (<340 K). Convective thermal diode walls where the heat flow is more intense in one direction than in the other has been studied in [5–8] where rectification factors as high as 8 have been reported. The idea of heat conduction thermal rectifier in a composite wall consisting of two materials, each having temperature dependant conductivity was presented in [9,10]. Some experimental results have also been discussed [11–13]. More recently, it has been shown [14] that the maximum rectification factor in a composite wall consisting of two solid materials, each having thermal conductivities with different temperature linear dependences, is equal to 3. In this paper we focus on steady coupled heat conduction and radiation problem within a one dimensional wall composed of two different materials. The first material is radiative and conductive while the second is only conductive and each thermal conductivity
1249
H. Sadat, V. Le Dez / Applied Thermal Engineering 107 (2016) 1248–1252
Nomenclature k1, k2 qt q+, q S T Tc Tm R x q, r, s t, u
thermal conductivities of the two considered media (W m1 K1) total heat flux in the medium (W m2) heat flux of the forward and reverse cases (W m2) volumic radiative source (W m3) temperature field in the medium (K) temperature of the interface between medium 1 and 2 (K) normalization temperature (K) rectification factor coordinate axis direction dimensioned coefficients of the forward and reverse 4th degree equations of the interface temperature solutions of the forward and reverse resolvant 3rd degree equations
is constant. We shall assume that radiative transfer can be described by the Rosseland approximation and therefore has a power three increase of its radiative conductivity with temperature. This variation of the total conductivity of the first wall with temperature let us expect a rectification factor of the order of that obtained in pure heat conduction problems. In the following sections, the problem is solved analytically and an expression for the rectification factor is given. It is shown that the rectification factor has a maximum value of 2.6 which is indeed of the order of what is obtained in pure heat conduction problems. In order to examine the influence of the extinction coefficient, we have also considered the limiting case of an optically thin media where only surface radiation is taken into account in the first layer. Here again, it is found that rectification factor is bounded by the same maximum value. 2. Heat conduction model and radiation Rosseland approximation Let us consider a one-dimensional heat exchange problem with two solid different materials as depicted in Fig. 1. The first domain is a conductive hot emitting and highly absorbing grey semitransparent medium, characterized by its thermal conductivity k1 and its absorption coefficient j, while the other domain is made of an opaque conductive medium characterized by its thermal conductivity k2 only. The non-dimensional width of each of them is 0.5 giving a total thickness of 1. The thermal conductivities k1 and k2 of the two materials are both supposed to be constant. The first material is a participating medium with a high absorption coefficient, for which radiative transfer within it can be described by the Rosseland approximation. This approximation stands that the radiative transfer can be described as a conductive one, with an apparent thermal conductivity depending both on the absorption coefficient and the temperature. Its total conductivity can therer T 3 is the fore be written as: kðTÞ ¼ k1 þ kR ðTÞ, where kR ðTÞ ¼ 16 3j Rosseland conductivity. Without loss of generality we will suppose that the boundary conditions are T(0) = T1 and T(1) = 0. The problem with an imposed non zero temperature at x = 1 can be retrieved by a variable change. These boundary conditions which
y, z
non dimensional solutions of the forward and reverse resolvant equations
Greek letters a, b, c non-dimensional coefficients analogous to q, r, s j absorption coefficient (m1) k total conductivity (W m1 K1) r Stephan-Boltzmann constant (5.67 108 W m2 K4) q, s non dimensional variables defining the rectification factor q1, qc reflection factors of the boundary surfaces of medium 1 e1, ec emissivities of the boundary surfaces of medium 1
correspond to the forward case are inverted in the reverse case (T(0) = 0 and T(1) = T1). In all what follows, one notes k for the total conductivity: in the opaque medium 2 it reduces only to the thermal conductivity k2, while in the radiatively participating medium 2 it is the sum of the thermal conductivity k1 and the Rosseland conductivity kR(T). The heat equation writes everywhere in the slab:
d dT ¼0 k dx dx
ð1Þ
The solution is simply given by k dT ¼ b where b is a constant dx depending on the boundary conditions. 2.1. Forward case In the second domain x 2 12 ; 1 where k ¼ k2 , the differential equation for the temperature field has the simple general solution T ¼ kb2 x þ c, c being another constant to be determined, associated to the two boundary conditions T(1) = 0 and T 12 ¼ T c . The determination of the two unknown constants easily gives the heat flux field in the domain: qt = 2k2Tc. In the first region: x 2 0; 12 , we can write:
k
dT d 4r 4 T ¼ k1 Tþ dx dx 3jk1 The
heat
equation
has
ð2Þ therefore
the
general
solution
T þ 34jrk1 T 4 ¼ kb1 x þ c. The two boundary conditions T(0) = T1 and T 12 ¼ T c allow the determination of the two constants, which leads to the heat flux:
4r 3 4r 3 qt ¼ 2k1 T c 1 þ Tc T1 1 þ T1 3jk1 3jk1
ð3Þ
The continuity of the heat flux at x ¼ 12 gives the algebraic equation verified by the interface temperature:
T 4c þ
3jk1 k2 3jk1 T 1 4r 3 Tc 1þ 1þ T1 ¼ 0 4r k1 4r 3jk1
Fig. 1. Heat conduction problem.
ð4Þ
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This 4th degree equation, of general form T 4c ¼ qT c þ r, with
q ¼ 34jrk1 1 þ kk21 and r ¼ 3j4kr1 T 1 1 þ 34jrk1 T 31 , can exactly be solved by the Ferrari’s method, and the temperature Tc at the interface between the two media can be expressed as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffi 1 2jqj pffiffi t t Tc ¼ 2 t
where t ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 q2 þ q4 þ4ð4r 3Þ 2
sq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3 3 q4 þ4ð4r q2 3Þ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffi 2jqj pffiffi t t t
s
ð5Þ
Ts is a reference temperature, and that the temperature Tm can be seen as the reference temperature for which the corresponding Stark number equals 13. With these notations, the coefficients of the 4th degree interface temperature equations for the forward and reverse cases define 3
> 0 is the solution of the
non-dimensional coefficients a, b and c, with jqj ¼ aT 3m , r ¼ bT 4m
third degree associated resolvant equation t3 + 4rt q2 = 0. Then the heat flux of the forward case is simply:
qþ ¼ k2
Note that for a steady-state conduction-radiation transfer inside a semi-transparent grey medium of absorption coefficient j and thermal conductivity k, one may introduce a conductionradiation parameter, the Stark number, defined by N S ¼ 4rjTk 3 , where
ð6Þ
and s ¼ cT 4m , where a = 1 + q, b = s(1 + s3) and c = qs. Introducing sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 a2 þ a4 þ4ð4b 3Þ the two supplementary quantities: z¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sq sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 3 3 3 3 a4 þ4ð4b a2 a2 þ a4 þ4ð43cÞ a4 þ4ð43cÞ a2 3Þ and y ¼ , with 2 2 2 z ¼ Tt2 and y ¼ Tu2 , which are the non-dimensional Cardan solutions m
2.2. Reverse case In the reverse case, the general solution obtained in the forward case remains valid, with T ¼ kb2 x þ c for x 2 12 ; 1 and 0 T þ 34jrk1 T 4 ¼ kb1 x þ c0 for x 2 0; 12 . Application of the boundary conditions sets leads to the two expressions of the heat flux as qt = 2k2(T1 Tc) in the 2nd region
and qt ¼ 2k1 T c 1 þ 34jrk1 T 3c in the first one. Similarly to what precedes, the heat flux continuity gives the algebraic 4th degree equation verified by the interface temperature under the form:
T 4c þ
3jk1 k2 3jk2 T 1 Tc 1þ ¼0 4r k1 4r
ð7Þ
Defining the parameter s ¼ 3j4kr2 T 1 gives the general form of the 4th degree equation for the interface temperature T 4c ¼ qT c þ s, where the parameter q has the same expression as in the forward case, and the solution u of the associated resolvant third degree sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 3 3 q2 þ q4 þ4ð4s q4 þ4ð4s q2 3Þ 3Þ > 0. The temperature equation: u ¼ 2 2 Tc at the interface between the two media and the heat flux of the reverse case are finally given by the following expressions:
1 Tc ¼ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi 2jqj pffiffiffi u u u
ð8Þ
q ¼ k2
qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2affi p z z z Rðq; sÞ ¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2s þ y p2affiffiy y
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi 2jqj pffiffiffi u u 2T 1 u
ð9Þ
2.3. Rectification factor The steady state heat fluxes in the forward and the reverse cases given by relations (6) and (9) allow us to define the rectification
þ
factor as the ratio R ¼ qq . This factor is given by the following
ð11Þ
It is possible to show by taking the limit of the previous expression that R ? 1 for s ? 0. To obtain this result, a limited develop qffiffiffiffiffiffiffiffiffiffiffiffi 4 1 2affi p zs!0 a3 1 þ 4a3 3 s , ment in the variable s shows that z qffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffi 4 1 1 3 2affiffi p zs!0 a3 1 2a3 3 s , ys!0 a3 1 þ 4qa3 s and y q ffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffi pffiffiffi 1 3 ys!0 a3 1 2qa3 s , whence 2s þ y p2affiffiy ys!0 qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2affi 2s 1 qa ¼ 2as and p z zs!0 2as for any q. z
We present in Fig. 2 the rectification factor as a function of q and s. The first map on Fig. 2a shows the general representation of the rectification factor for a large spectrum of the two parameters q and s, while the Fig. 2b focuses on a smaller region of small s where the rectification factor takes significant values >1. As it can be observed, there exists a thin band whose thickness increases with r where R has high values. The maximum value of the rectification factor is numerically seen to be around: Rm 2.62. The observation of the numerical results shows that if q P 3 then
R P 2 if 0:73q3 6 s 6 1:47q3 . In particular, it can be seen that R 6 1:5 as soon as s P 15 for reasonable q, and that R ? 1 for s ? + 1 for any q. As a matter of example, one can see that a value of the rectification factor R around 2 can be reached only for s P 1:3 which implies that the chosen temperature T1 must be high enough if the absorption coefficient is not too small: if one chooses T1 = 1000 K, one obtains Tm 770 K and jk1 34:5. This case corresponds to a high absorption coefficient and a small thermal conductivity k1 with a large spectrum of conductivity of the second layer: k2 P 2k1 . 1
and
m
of the two resolvant third degree equations of the forward and reverse cases, allows writing the rectification factor as a function of the two independent non-dimensional parameters q and s under the form:
1
formula:
qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi 2jqj pffi t t t R¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi u 2T 1 þ u 2jqj u
ð10Þ
Let us define a normalization temperature as T 3m ¼ 34jrk1 which appears in Eqs. (4) and (7), and the two non-dimensional parameters
s ¼ TTm1 and q ¼ kk21 .
3. Heat conduction model and thin media radiation approximation In the case of a thin radiation approximation with an absorption coefficient tending towards 0 (surface radiation), conservation of energy in medium 1 leads to write that the total heat flux qt = qr + qc, sum of the radiative flux and the conductive one, is constant across the medium. This implies that:
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Fig. 2. Rectification factor versus the two parameters q and s.
re1 ec ðT 41 T 4c Þ dT ¼x k1 1 q1 qc dx
ð12Þ
where x is a constant depending on the boundary conditions.
T 4c þ
3.1. Forward case In the first domain x 2 0; 12 , Eq. (12) can be solved as follows:
1 TðxÞ ¼ k1
"
#
re1 ec ðT 4i T 4c Þ x xþb 1 q1 qc
ð13Þ
where b is a constant to be determined. The boundary conditions, re e ðT 4 T 4 Þ T 12 ¼ T c and T(0) = T1, easily lead to: x ¼ 11c q 1q c 2k1 ðT c T 1 Þ, 1
c
and the total flux in domain 1 writes:
qt ¼
re e
4 1 c ðT 1
T 4c Þ
1 q1 qc
2k1 ðT c T 1 Þ
ð14Þ
There is no modification to the previous case in the second domain x 2 12 ; 1 where the total flux consists only in a conductive one, given by qt = 2k2Tc. Continuity of the heat flux at the interface leads to the 4th degree equation verified by the temperature Tc:
k2 Tc 1þ e1 ec r k1 2k1 ð1 q1 qc ÞT 1 e1 ec r 1þ T 31 ¼ 0 e1 ec r 2k1 ð1 q1 qc Þ
T 4c þ
2k1 ð1 q1 qc Þ
ð15Þ
which is exactly similar to the equation obtained in the Rosseland approximation, with the substitution:
q1 qc Þ j $ 8ð1 . 3e1 ec
the
first
domain
x 2 0; 12 ,
the
total
flux
writes:
x ¼ r1e1qe1c qT cc k1 dT , and the temperature variation is given by: dx 4
! 1 re1 ec T 4c TðxÞ ¼ þx xþb k1 1 q1 qc The boundary conditions, T total flux expression:
qt ¼ 2k1 T c 1 þ
2k1 ð1 q1 qc Þ
e1 ec r
k2 2k2 ð1 q1 qc ÞT 1 Tc 1þ ¼0 k1 e1 ec r
The same substitution
ð18Þ
q1 qc Þ j $ 8ð1 allows to retrieve the equa3e1 ec
tion verified in the Rosseland approximation. q1 qc Þ Then introducing the normalization temperature T 3m ¼ 2k1 ð1 e1 ec r ,
so as the independent parameters s ¼ TTm1 and q ¼ kk21 , allows writing the rectification factor as:
qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2affiffi p z z x R¼ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 2s þ y p2affiffiy y
ð19Þ
where a, z and y have exactly the same expressions as in the Rosseland approximation frame. One concludes that the map representing the rectification factor for the thin media approximation is identical to the one obtained for the Rosseland approximation, and that the maximal value of the rectification factor is Rm 2.62. In the case of black boundaries, the normalization temperature is: T 3m ¼ 2kr1 . For a thin medium with k1 = 1 W m1 K1, this leads to Tm 328 K and s 3 when T1 = 1000 K and relatively high values of R around 2 can be obtained for media with k2 P 2, the maximum Rm being reached for k2 30. If k1 ¼ 0:1 W m1 K1 , Tm 152 K and s 6.6, leading to the maximal value of R reached for media with k2 P 18. 4. Conclusion
3.2. Reverse case In
Similarly there is no modification in the area 12 6 x 6 1 where the flux writes qt = 2k2(T1 Tc). The equation verified by the interface temperature Tc is:
1 2
e1 ec r T3 2k1 ð1 q1 qc Þ c
ð16Þ ¼ T c and (0) = 0, now lead to the
ð17Þ
In this study we developed an analytical expression for the rectification factor in a bilayer wall when radiation and conduction are coupled in the first layer. By using the Rosseland approximation and by considering surface radiation in the first layer where conduction is also taken into account while the second layer is only conductive, we have given an analytical expression for the rectification factor. A theoretical maximum value has been numerically found, which can hardly be obtained in the Rosseland approximation frame, except for a very small thermal conductivity in the hot domain. On the contrary for the thin media approximation, values close to the theoretical maximum rectification factor can be obtained in less restrictive conditions.
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